Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Elliptic three-folds II: Multiple fibres

Author: Mark Gross
Journal: Trans. Amer. Math. Soc. 349 (1997), 3409-3468
MSC (1991): Primary 14J30
MathSciNet review: 1401771
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $f:X\rightarrow S$ be an elliptic fibration with a section, where $S$ is a projective surface and $X$ is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset $Z\subseteq S$ to obtain a new elliptic fibration $f':X'\rightarrow S$ which now has multiple fibres along $Z$. This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of $X$.

References [Enhancements On Off] (What's this?)

  • 1. Barth, W., Peters, C., and Van de Ven, A., Compact Complex Surfaces, (Ergeb. Math. Grenzbeg. 3. Folge, vol. 4), Springer, 1984. MR 86c:32026
  • 2. Bloch, S., Torsion Algebraic Cycles and a Theorem of Roitman, Compositio Math. 39 (1979), 107-127. MR 80k:14012
  • 3. Bloch, S., and Ogus, A., Gersten's Conjecture and the Homology of Schemes, Ann. Éc. Norm. Sup. 7 (1974), 181-202. MR 54:318
  • 4. Cossec, F., and Dolgachev, I., Enriques Surfaces I, Birkhäuser, Boston, 1989. MR 90h:14052
  • 5. Colliot-Thélène, J.L., Sansuc, J.J., and Soulé, C., Torsion dans le Groupe de Chow de Codimension Deux, Duke Math. J. 50 (1983), 763-801. MR 85d:14010
  • 6. Cox, D., and Zucker, S., Intersection Numbers of Sections of Elliptic Surfaces, Inventiones Math. 53 (1979), 1-44. MR 81i:14023
  • 7. Deligne, P., Courbes Elliptiques: Formulaire d'après J. Tate, Modular Functions in One Variable IV, Lecture Notes in Mathematics, vol. 476, Springer-Verlag, 1975, pp. 53-74. MR 52:8135
  • 8. Dolgachev, I., and Gross, M., Elliptic Three-folds I: Ogg-Shafarevich Theory, J. Algebraic Geometry 3 (1994), 39-80. MR 95d:14037
  • 9. Fujimoto, Y., Logarithmic Transformations on Elliptic Fibre Spaces, J. Math. Kyoto Univ. 28 (1988), 91-110. MR 88m:32053
  • 10. Fulton, W., Intersection Theory, (Ergeb. Math. Grenzgeb., 3. Folge, vol. 2), Berlin, Heidelberg, New York, Springer, 1984. MR 85k:14004
  • 11. Grassi, A., Minimal Models of Elliptic Threefolds, Ph.D. Thesis, Duke University, 1990.
  • 12. Gross, M., A Finiteness Theorem for Elliptic Calabi-Yau Threefolds, Duke Math. J. 74 (1994), 271-299. MR 95c:14047
  • 13. Grothendieck, A., Le Groupe de Brauer, I, II, III, Dix Exposés sur Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 46-188. MR 39:5586a,b,c
  • 14. Kodaira, K., On Compact Complex Analytic Surfaces, I, Ann. Math. 71 (1960), 111-152;
    II, Ann. Math 77 (1963), 563-626; III, Ann. Math. 78 (1963), 1-40. MR 24:A2396; MR 32:1730
  • 15. Milne, J., Étale Cohomology, Princeton Univ. Press, 1980. MR 81j:14002
  • 16. Miranda, R., Smooth Models for Elliptic Threefolds, Birational Geometry of Degenerations, Birkhäuser, 1983, pp. 85-133. MR 84f:14024
  • 17. Mumford, D., and Suominen, K., Introduction to the Theory of Moduli, Algebraic Geometry, Oslo 1970, Wolters-Noordhoff Press, 1972, pp. 171-222. MR 55:10455
  • 18. Nakayama, N., On Weierstrass Models, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata (1987), 405-431. MR 90m:14030
  • 19. Nakayama, N., Local Structure of an Elliptic Fibration, Preprint, Univ. of Tokyo, 1991.
  • 20. Ogg, A., Cohomology of Abelian Varieties over Function Fields, Ann. Math. 76 (1962), 185-212. MR 27:5758
  • 21. Shafarevich, I., Principal Homogeneous Spaces over Function Fields, AMS Translations 37 (1964), 85-113. MR 29:110 (Russian original)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 14J30

Retrieve articles in all journals with MSC (1991): 14J30

Additional Information

Mark Gross
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

Received by editor(s): June 19, 1995
Additional Notes: This material is based upon work supported by the North Atlantic Treaty Organization under a Grant awarded in 1990. Research at MSRI supported in part by NSF grant #DMS 9022140.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society